Table of Contents
Power Set Definition with Examples
A power set is a set of all sets that are subsets of a given set. In other words, a power set is the collection of all possible subsets of a set.
For example, the power set of the set {1, 2, 3} is the following set:
{ {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }
This set contains six sets: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}.
Define Power Set with Example
A power set is a set of all subsets of a given set.
For example, the power set of the set {1, 2, 3} would be:
{1, 2, 3}
{1, 2}
{1, 3}
{2, 3}
{1}
{2}
{3}
This is because the set {1, 2, 3} has six subsets: {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, and {2}.
Subsets
A subset is a group of elements that are part of a larger set. The smaller set is a part of the larger set, and can be identified by a number or letter. For example, the set {1, 2, 3, 4} has the subset {1, 2, 3}.
Number of Subsets
A formula to find the total number of subsets for a given set is the product of the set’s cardinality (the number of elements in the set) and the factorial of the number of elements minus 1. For example, the set {1, 2, 3} has a cardinality of 3 and a factorial of 3-1=2, so the total number of subsets for this set is 3×2=6.
Properties of Power Set
The power set of a set A is the set of all subsets of A.
The power set of a set A is always a subset of the set of all subsets of the power set of A.
The power set of a set A is a set of cardinality 2n, where n is the number of elements in A.
Cardinality of a Power Set
The cardinality of a power set is the number of possible combinations of elements in a set. For a set with n elements, the cardinality of the power set is 2^n.
Power Set of Empty Set
The empty set is a set with no elements.
Power Set of a Countable Set
The power set of a countable set is the set of all subsets of the given set.
Power Set of an Uncountable Set
There is no set of all possible sets.
